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Casimir operator in su(2)

by hjlim
Tags: casimir, su2
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Mar1-10, 08:06 PM
P: 5
Hi all,

If I define Tij = a+i aj, then

C2 = T11T11 + T12T21 + T21T12 + T22T22 is a second order casimir operator.

For SU(2), it's [tex]\frac{N}{2}[/tex] ([tex]\frac{N}{2}[/tex] + 1)

But as I calculate it directly,
C2 = a+1 a1a+1 a1 + a+1 a2a+2 a1 + a+2 a1a+1 a1 + a+2 a2a+2 a2 =
a+1 a1a+1 a1 + a+1 (a+2a2 + 1)a1 + a+2 (a+1a1 + 1)a2 + a+2 a2a+2 a2 =
N1N1 + N1(N2 + 1) + N2(N1 + 1) + N2N2 = (N1 + N2)2 + N1 + N2 = N(N + 1)

which is different from above. Can you let me know what is wrong with my argument? Thank you very much!
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